Should I be asked for my password when a bug report is sent?

Sometimes, but not usually.

Reporting Crashes in System Processes

Most of the time when an application crashes or you report a bug manually, you don't have to put in a password for information to be automatically gathered and sent to Launchpad.

However, when a program or service that runs as root crashes, collecting data about that crash requires accessing data that is only accessible as root. In Ubuntu, you can perform actions as root by authenticating with your password, provided that your account has administrative capabilities. (This is facilitated by sudo or one of its graphical frontends, or by PolicyKit.) If your account does not have administrative capabilities, then you will not be prompted to report a crash in a root-owned process until you log in with an account that has administrative capabilities.

When you're logged in as an administrator and you're prompted to report a crash in a root-owned process, you see a password dialog like this:

This is one of the two situations where you may have to enter a password to report a bug in Ubuntu.

Once you've authenticated, then the crash-reporting process continues the same way as for regular non-root processes.

Filing Bug Reports for Crashes

Before Ubuntu 12.04 LTS, when Apport crash reporting was enabled, you would be asked to submit a bug report to contain the data gathered automatically. This is still what happens in alpha and beta versions of Ubuntu (and versions before 12.04 LTS). But starting with Ubuntu 12.04 LTS, Apport is set up by default so that, if you are running a stable version of Ubuntu and a crash occurs, it will automatically (with your permission) submit the information to a database that doesn't require you to fill out a bug report. This is potentially less useful than having a bug report, but it means that information can effectively be gathered from many more users--plus, bug reports are often most valuable for alpha and beta releases where the bugs are more likely to be fixed as a result of the report.

However, if you re-enable bug filing for crashes, or you report a bug by invoking ubuntu-bug manually with a package name or process ID, then you will be given the opportunity to file a bug report. Bug reports in Ubuntu are filed on Launchpad. So once data for the bug are collected and submitted to Launchpad, a Launchpad page to file the bug report with those data automatically attached will appear in a web browser. (Unless your default web browser doesn't work and that's what the bug is about, of course.)

If you're not logged in to Launchpad, then you'll be brought to the Launchpad sign-in page in your web browser, to report the bug. This is the second of the two situations where you may have to enter a password, to report a bug in Ubuntu.